Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence

Jacob B. Khurgin; Greg Sun

doi:10.1364/OE.20.015309

Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence

Jacob B. Khurgin and Greg Sun

Optics Express
Vol. 20,
Issue 14,
pp. 15309-15325
(2012)
•https://doi.org/10.1364/OE.20.015309

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Jacob B. Khurgin and Greg Sun, "Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence," Opt. Express 20, 15309-15325 (2012)

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Related Topics
Table of Contents Category
- Optics at Surfaces
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Surface plasmons (240.6680)
- Plasmonics (250.5403)

About this Article
History
- Original Manuscript: May 1, 2012
- Manuscript Accepted: June 1, 2012
- Published: June 22, 2012

Accessible

Open Access

Abstract

We develop a theoretical model for the semiconductor generator of the sub-wavelength surface plasmons, operating on a single mode and often referred to as a spaser. We show that input-output characteristics of the single mode spaser does not exhibit nonlinearity inherent in most lasers, but the linewidth of the emission collapses, as in any laser which allows us to define the threshold. Our rigorous derivations show that as long as the mode remains substantially sub-wavelength in all three dimensions, the threshold current (power) shows virtually no dependence on the gain material and geometry of the active layer and is determined solely by the intrinsic loss of the metal in the device. For the semiconductor single mode surface plasmon generators operating in the telecommunication range the threshold current is on the order of 10-20 µA, and the threshold current density grows fast with the decrease of the device size reaching 100’s of kA/cm² or more. This fact makes coherent sources of sub-wavelength SP’s unattainable from our point of view, but there exists a room for efficient broad-band incoherent SP sources either optically or electrically pumped.

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References

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Publication

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992).
[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008).
[Crossref]
M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
[Crossref]
M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011).
[Crossref] [PubMed]
M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004).
[Crossref] [PubMed]
C.-Y. Lu and S. L. Chuang, “A surface-emitting 3D metal-nanocavity laser: proposal and theory,” Opt. Express 19(14), 13225–13244 (2011).
[Crossref] [PubMed]
M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]
R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]
M. T. Hill, M. Marell, E. S. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[Crossref] [PubMed]
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[Crossref]
S.-H. Kwon, J.-H. Kang, C. Seassal, S.-K. Kim, P. Regreny, Y.-H. Lee, C. M. Lieber, and H.-G. Park, “Subwavelength plasmonic lasing from a semiconductor nanodisk with silver nanopan cavity,” Nano Lett. 10(9), 3679–3683 (2010).
[Crossref] [PubMed]
A. M. Lakhani, M.-K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
[Crossref] [PubMed]
J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, “Electrically pumped sub-wavelength metallo-dielectric pedestal pillar lasers,” Opt. Express 19(22), 21524–21531 (2011).
[Crossref] [PubMed]
M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010).
[Crossref]
M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]
R. F. Oulton, “Plasmonics: Loss and gain,” Nat. Photonics 6(4), 219–221 (2012).
[Crossref]
R. F. Oulton, “Surface plasmon lasers: sources of nanoscopic light,” Mater. Today 15(1-2), 26–34 (2012).
[Crossref]
R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).
P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6(1), 16–24 (2011).
[Crossref]
F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006).
[Crossref] [PubMed]
J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]
P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]
J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]
M. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys. 10(10), 105006 (2008).
[Crossref]
A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[Crossref]
E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids 1(4), 249–261 (1957).
[Crossref]
L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).
I. Vurgaftman, J. R. Meyer, and L.-R. Ram-Mohan, “Band parameters for III–V compound semiconductors and their alloys,” J. Appl. Phys. 89(11), 5815–5875 (2001).
[Crossref]
G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42(4), 799–810 (1995).
[Crossref]
M. J. Connelly, Semiconductor Optical Amplifiers (Springer-Verlag, 2002).
A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]
H. Statz, and G. De Mars, Quantum Electronics, ed. C. H. Townes (Columbia University Press, 1960) 530.
D. A. Kleiman, “The maser rate equations and spiking,” Bell Syst. Tech. J. 43, 1505–1532 (1964).
G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50(2), 1675–1680 (1994).
[Crossref] [PubMed]
R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission From GaAs Junctions,” Phys. Rev. Lett. 9(9), 366–368 (1962).
[Crossref]
I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970).
[Crossref]
Zh. I. Alferov and R. F. Kazarinov, “Semiconductor laser with electric pumping,” Inventor’s Certificate No. 181737 [in Russian], Application No. 950840, priority as of March 30, 1963.
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[Crossref]
J. B. Khurgin and G. Sun, “In search of the elusive lossless metal,” Appl. Phys. Lett. 96(18), 181102 (2010).
[Crossref]
G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009).
[Crossref]
G. Sun and J. B. Khurgin, “Plasmon enhancement of luminescence by metal nanoparticles,” IEEE J. Sel. Top. Quantum Electron. 17(1), 110–118 (2011).
[Crossref]

2012 (6)

K. Ding, Z. C. Liu, L. J. Yin, M. T. Hill, M. J. H. Marell, P. J. van Veldhoven, R. Nöetzel, and C. Z. Ning, “Room-temperature continuous wave lasing in deep-subwavelength metallic cavities under electrical injection,” Phys. Rev. B 85(4), 041301 (2012).
[Crossref]

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

R. F. Oulton, “Plasmonics: Loss and gain,” Nat. Photonics 6(4), 219–221 (2012).
[Crossref]

R. F. Oulton, “Surface plasmon lasers: sources of nanoscopic light,” Mater. Today 15(1-2), 26–34 (2012).
[Crossref]

R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]

2011 (7)

A. M. Lakhani, M.-K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
[Crossref] [PubMed]

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, “Electrically pumped sub-wavelength metallo-dielectric pedestal pillar lasers,” Opt. Express 19(22), 21524–21531 (2011).
[Crossref] [PubMed]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6(1), 16–24 (2011).
[Crossref]

M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011).
[Crossref] [PubMed]

C.-Y. Lu and S. L. Chuang, “A surface-emitting 3D metal-nanocavity laser: proposal and theory,” Opt. Express 19(14), 13225–13244 (2011).
[Crossref] [PubMed]

G. Sun and J. B. Khurgin, “Plasmon enhancement of luminescence by metal nanoparticles,” IEEE J. Sel. Top. Quantum Electron. 17(1), 110–118 (2011).
[Crossref]

2010 (4)

J. B. Khurgin and G. Sun, “In search of the elusive lossless metal,” Appl. Phys. Lett. 96(18), 181102 (2010).
[Crossref]

S.-H. Kwon, J.-H. Kang, C. Seassal, S.-K. Kim, P. Regreny, Y.-H. Lee, C. M. Lieber, and H.-G. Park, “Subwavelength plasmonic lasing from a semiconductor nanodisk with silver nanopan cavity,” Nano Lett. 10(9), 3679–3683 (2010).
[Crossref] [PubMed]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010).
[Crossref]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
[Crossref]

2009 (4)

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

M. T. Hill, M. Marell, E. S. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[Crossref] [PubMed]

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009).
[Crossref]

2008 (2)

M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008).
[Crossref]

M. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys. 10(10), 105006 (2008).
[Crossref]

2006 (1)

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006).
[Crossref] [PubMed]

2005 (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[Crossref]

2004 (1)

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004).
[Crossref] [PubMed]

2003 (1)

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[Crossref] [PubMed]

2001 (1)

I. Vurgaftman, J. R. Meyer, and L.-R. Ram-Mohan, “Band parameters for III–V compound semiconductors and their alloys,” J. Appl. Phys. 89(11), 5815–5875 (2001).
[Crossref]

1999 (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[Crossref] [PubMed]

1995 (1)

G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42(4), 799–810 (1995).
[Crossref]

1994 (1)

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50(2), 1675–1680 (1994).
[Crossref] [PubMed]

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992).
[Crossref]

1972 (1)

P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

1970 (1)

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970).
[Crossref]

1964 (1)

D. A. Kleiman, “The maser rate equations and spiking,” Bell Syst. Tech. J. 43, 1505–1532 (1964).

1963 (1)

H. Kroemer, “A proposed class of heterojunction injection lasers,” Proc. IEEE 51(12), 1782–1783 (1963).
[Crossref]

1962 (1)

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission From GaAs Junctions,” Phys. Rev. Lett. 9(9), 366–368 (1962).
[Crossref]

1958 (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]

1957 (1)

E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids 1(4), 249–261 (1957).
[Crossref]

Bakker, R.

Bartal, G.

Belgrave, A. M.

Bergman, D. J.

Berini, P.

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6(1), 16–24 (2011).
[Crossref]

Björk, G.

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50(2), 1675–1680 (1994).
[Crossref] [PubMed]

Bondarenko, O.

Carlson, R. O.

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission From GaAs Junctions,” Phys. Rev. Lett. 9(9), 366–368 (1962).
[Crossref]

Christy, R.

P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Chuang, S. L.

C.-Y. Lu and S. L. Chuang, “A surface-emitting 3D metal-nanocavity laser: proposal and theory,” Opt. Express 19(14), 13225–13244 (2011).
[Crossref] [PubMed]

Dai, L.

Dapkus, P. D.

De Leon, I.

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6(1), 16–24 (2011).
[Crossref]

Ding, K.

Fainman, Y.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004).
[Crossref] [PubMed]

Feng, L.

Fenner, G. E.

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission From GaAs Junctions,” Phys. Rev. Lett. 9(9), 366–368 (1962).
[Crossref]

Foy, P. W.

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970).
[Crossref]

Geluk, E. J.

Gladden, C.

Gu, Q.

Halas, N. J.

Hall, R. N.

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission From GaAs Junctions,” Phys. Rev. Lett. 9(9), 366–368 (1962).
[Crossref]

Hayashi, I.

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970).
[Crossref]

Herz, E.

Hill, M. T.

Johnson, P. B.

P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Kane, E. O.

E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids 1(4), 249–261 (1957).
[Crossref]

Kang, J.-H.

Karlsson, A.

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50(2), 1675–1680 (1994).
[Crossref] [PubMed]

Karouta, F.

Katz, M.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

Khajavikhan, M.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

Khurgin, J. B.

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]

G. Sun and J. B. Khurgin, “Plasmon enhancement of luminescence by metal nanoparticles,” IEEE J. Sel. Top. Quantum Electron. 17(1), 110–118 (2011).
[Crossref]

J. B. Khurgin and G. Sun, “In search of the elusive lossless metal,” Appl. Phys. Lett. 96(18), 181102 (2010).
[Crossref]

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009).
[Crossref]

Kim, I.

Kim, M.-K.

A. M. Lakhani, M.-K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
[Crossref] [PubMed]

Kim, S.-K.

Kingsley, J. D.

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission From GaAs Junctions,” Phys. Rev. Lett. 9(9), 366–368 (1962).
[Crossref]

Kleiman, D. A.

D. A. Kleiman, “The maser rate equations and spiking,” Bell Syst. Tech. J. 43, 1505–1532 (1964).

Knight, M.

Kroemer, H.

H. Kroemer, “A proposed class of heterojunction injection lasers,” Proc. IEEE 51(12), 1782–1783 (1963).
[Crossref]

Kwon, S.-H.

Lakhani, A. M.

A. M. Lakhani, M.-K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
[Crossref] [PubMed]

Lau, E. K.

A. M. Lakhani, M.-K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
[Crossref] [PubMed]

Lee, J. H.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

Lee, R. K.

Lee, Y.-H.

Leong, E. S.

Levi, A. F. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992).
[Crossref]

Lieber, C. M.

Liu, Z. C.

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992).
[Crossref]

Lomakin, V.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

Lu, C.-Y.

C.-Y. Lu and S. L. Chuang, “A surface-emitting 3D metal-nanocavity laser: proposal and theory,” Opt. Express 19(14), 13225–13244 (2011).
[Crossref] [PubMed]

Ma, R.-M.

R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[Crossref]

Marell, M.

Marell, M. J. H.

McCall, S. L.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992).
[Crossref]

Meyer, J. R.

I. Vurgaftman, J. R. Meyer, and L.-R. Ram-Mohan, “Band parameters for III–V compound semiconductors and their alloys,” J. Appl. Phys. 89(11), 5815–5875 (2001).
[Crossref]

Mizrahi, A.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

Narimanov, E. E.

New, G. H. C.

G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42(4), 799–810 (1995).
[Crossref]

Nezhad, M. P.

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004).
[Crossref] [PubMed]

Ning, C. Z.

Nöetzel, R.

Noginov, M. A.

Nötzel, R.

O’Brien, J. D.

Oei, Y. S.

Oulton, R. F.

R. F. Oulton, “Plasmonics: Loss and gain,” Nat. Photonics 6(4), 219–221 (2012).
[Crossref]

R. F. Oulton, “Surface plasmon lasers: sources of nanoscopic light,” Mater. Today 15(1-2), 26–34 (2012).
[Crossref]

Oulton, R.F.

R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

Painter, O.

Panish, M. B.

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970).
[Crossref]

Park, H.-G.

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992).
[Crossref]

Ram-Mohan, L.-R.

I. Vurgaftman, J. R. Meyer, and L.-R. Ram-Mohan, “Band parameters for III–V compound semiconductors and their alloys,” J. Appl. Phys. 89(11), 5815–5875 (2001).
[Crossref]

Regreny, P.

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]

Scherer, A.

Seassal, C.

Shalaev, V. M.

Shen, Y. R.

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006).
[Crossref] [PubMed]

Simic, A.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

Slusher, R. E.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992).
[Crossref]

Slutsky, B.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

Smalbrugge, B.

Smit, M. K.

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[Crossref]

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R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission From GaAs Junctions,” Phys. Rev. Lett. 9(9), 366–368 (1962).
[Crossref]

Soref, R. A.

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009).
[Crossref]

Sorger, V. J.

R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

Stockman, M. I.

M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011).
[Crossref] [PubMed]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
[Crossref]

M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008).
[Crossref]

Stout, S.

Sumski, S.

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970).
[Crossref]

Sun, G.

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]

G. Sun and J. B. Khurgin, “Plasmon enhancement of luminescence by metal nanoparticles,” IEEE J. Sel. Top. Quantum Electron. 17(1), 110–118 (2011).
[Crossref]

J. B. Khurgin and G. Sun, “In search of the elusive lossless metal,” Appl. Phys. Lett. 96(18), 181102 (2010).
[Crossref]

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009).
[Crossref]

Sun, M.

Suteewong, T.

Tetz, K.

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004).
[Crossref] [PubMed]

Townes, C. H.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]

van Veldhoven, P. J.

Vurgaftman, I.

I. Vurgaftman, J. R. Meyer, and L.-R. Ram-Mohan, “Band parameters for III–V compound semiconductors and their alloys,” J. Appl. Phys. 89(11), 5815–5875 (2001).
[Crossref]

Wang, F.

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006).
[Crossref] [PubMed]

Wiesner, U.

Wu, M. C.

A. M. Lakhani, M.-K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
[Crossref] [PubMed]

Yamamoto, Y.

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50(2), 1675–1680 (1994).
[Crossref] [PubMed]

Yariv, A.

Yin, L. J.

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[Crossref]

Zentgraf, T.

Zhang, X.

R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

Zhu, G.

Zhu, Y.

Appl. Phys. Lett. (6)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992).
[Crossref]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970).
[Crossref]

J. B. Khurgin and G. Sun, “In search of the elusive lossless metal,” Appl. Phys. Lett. 96(18), 181102 (2010).
[Crossref]

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009).
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IEEE J. Sel. Top. Quantum Electron. (1)

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[Crossref]

J. Appl. Phys. (1)

I. Vurgaftman, J. R. Meyer, and L.-R. Ram-Mohan, “Band parameters for III–V compound semiconductors and their alloys,” J. Appl. Phys. 89(11), 5815–5875 (2001).
[Crossref]

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G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42(4), 799–810 (1995).
[Crossref]

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M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
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R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

Mater. Today (1)

R. F. Oulton, “Surface plasmon lasers: sources of nanoscopic light,” Mater. Today 15(1-2), 26–34 (2012).
[Crossref]

Nano Lett. (1)

Nat. Photonics (4)

M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008).
[Crossref]

R. F. Oulton, “Plasmonics: Loss and gain,” Nat. Photonics 6(4), 219–221 (2012).
[Crossref]

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6(1), 16–24 (2011).
[Crossref]

Nature (3)

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

New J. Phys. (1)

Opt. Express (5)

A. M. Lakhani, M.-K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
[Crossref] [PubMed]

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004).
[Crossref] [PubMed]

C.-Y. Lu and S. L. Chuang, “A surface-emitting 3D metal-nanocavity laser: proposal and theory,” Opt. Express 19(14), 13225–13244 (2011).
[Crossref] [PubMed]

Phys. Rep. (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[Crossref]

Phys. Rev. (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]

Phys. Rev. A (1)

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50(2), 1675–1680 (1994).
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Phys. Rev. Lett. (4)

M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011).
[Crossref] [PubMed]

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006).
[Crossref] [PubMed]

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission From GaAs Junctions,” Phys. Rev. Lett. 9(9), 366–368 (1962).
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Fig. 1

Geometry of the injection pumped spaser.

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Fig. 2

Effective mode volume of (a) Ag and (b) Au spheroids in absolute and normalized units.

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Fig. 3

Decay rates of SP modes in (a) gold and (b) silver spheroids.

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Fig. 4

The thickness of active layer required to achieve specified value of gain confinement factor in (a) silver and (b) gold spheroid modes.

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Fig. 5

(a) SP generation rate (output) and effective modal loss (linewidth) vs. pump current of Au-based spaser with a = 30nm and no spacing layer between the metal and gain medium (b) Carrier density as a function of pump current in the Au-based spaser with a = 30nm and no spacing layer between the metal and gain medium.

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Fig. 6

Snapshots of material gain profile and modal gain spectral density for different pumping conditions indicated in Figs. 5: (a) – well below transparence (b) at transparency (c) at threshold (d) well above threshold.

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Fig. 7

(a) Number of SP’s per mode (b) recombination rates and modal gain vs. carrier density in of Au-based spaser with a = 30nm and no spacing between the metal and gain medium

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Fig. 8

(A) SP generation rate (output) and effective modal loss (linewidth) vs. pump current of Au-based spasers (a) a = 40nm and Г = 0.8 (no spacing), (b) a = 15 nm and Г = 0.8 (no spacing), c) a = 40 nm and Г = 0.4 (2 nm spacing), (d) a = 15 nm and Г = 0.4 (2 nm spacing). (B) Carrier density as a function of pump current in the Au-based spasers (a) a = 40nm and Г = 0.8 (no spacing), (b) a = 15 nm and Г = 0.8 (no spacing), (c) a = 40 nm and Г = 0.4 (2 nm spacing), (d) a = 15 nm and Г = 0.4 (2 nm spacing).

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Fig. 9

(A) SP emission rate (output) and effective modal loss (linewidth) vs. pump current of Ag-based spasers (a) a = 40nm and Г = 0.8 (no spacing), (b) a = 15 nm and Г = 0.8 (no spacing), (c) a = 40 nm and Г = 0.4 (2 nm spacing), (d) a = 15 nm and Г = 0.4 (2 nm spacing). (B) Carrier density as a function of pump current in the Ag-based spasers (a) a = 40nm and Г = 0.8 (no spacing), (b) a = 15 nm and Г = 0.8 (no spacing), (c) a = 40 nm and Г = 0.4 (2 nm spacing), (d) a = 15 nm and Г = 0.4 (2 nm spacing).

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Fig. 10

(A) Threshold current density for Au-based spaser vs. the size of half-axis a, (a) Г = 0.8 (no spacing), (b) Г = 0.4 (2 nm spacing). (B) Threshold current density for Ag-based spaser vs. the half-axis a, (a) Г = 0.8 (no spacing), (b) Г = 0.4 (2 nm spacing).

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Equations (32)

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\frac{1}{2} ε_{0} ε_{s} E_{m a x}^{2} V_{e f f} = U_{t}

U_{t} = \frac{1}{2} ε_{0} \int_{m e t}^{} E^{2} (r) d V + \frac{1}{2} ε_{0} ε_{s} \int_{S e m i}^{} E^{2} (r) d V

U_{m a g} = \frac{1}{4} μ_{0} \int_{m e t + s e m i}^{} H^{2} (r) d V

γ_{n r a d} = 2 γ \frac{U_{k i n}}{U_{t}} = γ (1 - 2 \frac{U_{m a g}}{U_{t}})

γ_{r a d} = \frac{2 n ω}{9} (1 + \frac{| ε_{m} |}{ε_{s}}) {(\frac{2 π}{λ})}^{3} a b^{2}

\frac{d n_{c} (r)}{d t} = - \frac{2 π}{ħ} \frac{1}{3} \frac{e^{2} P^{2}}{m_{0}^{2} ω^{2}} \frac{1}{2 π^{2}} {(\frac{2 μ_{r}}{ħ^{2}})}^{\frac{3}{2}} {(ħ ω - E_{g})}^{\frac{1}{2}} (f_{c} - f_{v}) E^{2} (r)

f_{c (v)} (E) = \frac{1}{1 + exp (\frac{E - μ_{c (v)}}{k T})}

n_{c} = \frac{1}{2 π^{2}} {(\frac{2 m_{e}}{ħ^{2}})}^{3 / 2} \int^{​} E_{c}^{1 / 2} f_{c} (E_{c}) d E_{c}

\frac{d N_{S P}}{d t} = - \frac{d N_{c}}{d t} = \frac{2 π}{ħ} \frac{1}{3} \frac{e^{2} P^{2}}{m_{0}^{2} ω^{2}} \frac{1}{2 π^{2}} {(\frac{2 μ_{r}}{ħ^{2}})}^{\frac{3}{2}} {(ħ ω - E_{g})}^{\frac{1}{2}} \frac{2 ħ ω N_{S P}}{ε_{0} ε_{s}} (f_{c} - f_{v}) Г = g (ω) N_{S P}

Г = \frac{\int_{g a i n} E^{2} (r) d V}{E_{m a x}^{2} V_{e f f}}

g (ω) = \frac{8 α_{0} c}{3 ε_{s}} R_{P} {[\frac{2 μ_{r}}{ħ^{2}} (ħ ω - E_{g})]}^{\frac{1}{2}} (f_{c} - f_{v}) Г = A k_{ω} (f_{c} - f_{v}) Г

k_{ω} = {[\frac{2 μ_{r}}{ħ^{2}} (ħ ω - E_{g})]}^{\frac{1}{2}}

γ_{m} = γ_{m 0} - g_{m} .

g_{m} = \int_{0}^{\infty} g (ω) \frac{(γ_{m 0} - g_{m}) / 2 π}{{(ω - ω_{0})}^{2} + {(γ_{m 0} - g_{m})}^{2} / 4} d ω

ρ_{ω} = \frac{2 n^{3} ω^{2}}{3 π^{2} c^{3}} .

d E_{f s}^{2} = \frac{4 n ω^{3}}{3 π^{2} ε_{0} c^{3}} d (ħ ω) .

τ_{f s, ω}^{- 1} = \frac{2 π}{ħ} \frac{1}{3} \frac{e^{2} P^{2}}{m_{0}^{2} ω^{2}} \frac{4 n ω^{3}}{3 π^{2} ε_{0} c^{3}} f_{c} (1 - f_{v}) = \frac{16}{9} α_{0} R_{P} \frac{ħ}{μ_{r}} \frac{n ω^{2}}{c^{2}} f_{c} (1 - f_{v})

\begin{matrix} r_{s p, f s} = \frac{V_{a}}{2 π^{2}} {(\frac{2 μ_{r}}{ħ^{2}})}^{3 / 2} \int^{​} \frac{1}{τ_{f s, ω}} {(ħ ω - E_{g})}^{1 / 2} d (ħ ω) \\ = \frac{64}{9} α_{0} R_{P} \frac{V_{a} n}{ħ λ^{2}} {(\frac{2 μ_{r}}{ħ^{2}})}^{1 / 2} \int^{​} f_{c} (1 - f_{v}) {(ħ ω - E_{g})}^{1 / 2} d (ħ ω) \\ = \frac{64}{9} α_{0} R_{P} \frac{V_{a} n}{ħ λ^{2}} \int^{​} f_{c} (1 - f_{v}) k_{ω} d (ħ ω) \end{matrix}

r_{s p, f s} = \frac{16 π}{3} \frac{V_{e f f}}{ω λ_{s}^{3} Г} \int^{​} g (ω) n_{s p} (ω) d ω

n_{s p} (ω) = \frac{f_{c} (1 - f_{v})}{(f_{c} - f_{v})} > 1

ρ_{ω} = \frac{E^{2} (r)}{E_{m a x}^{2} V_{e f f}} \frac{γ_{m} / 2 π}{{(ω - ω_{0})}^{2} + γ_{m}^{2} / 4}

\begin{array}{l} r_{s p, m} = \frac{1}{2 π^{2}} {(\frac{2 μ_{r}}{ħ^{2}})}^{3 / 2} \int^{​} \frac{1}{τ_{m, ω}} {(ħ ω - E_{g})}^{1 / 2} d (ħ ω) \\ = \frac{8}{3} α_{0} R_{P} \frac{c}{ħ ε_{s}} Г \int^{​} \frac{γ_{m} / 2 π}{{(ω - ω_{0})}^{2} + γ_{m}^{2} / 4} f_{c} (1 - f_{v}) k_{ω} d (ħ ω) . \end{array}

r_{s p, m} = \int^{​} g (ω) n_{s p} (ω) \frac{γ_{m} / 2 π}{{(ω - ω_{0})}^{2} + γ_{m}^{2} / 4} d ω = g_{m} {\bar{n}}_{s p, m}

{\bar{n}}_{s p, m} = \frac{\int^{​} g (ω) n_{s p} (ω) \frac{γ_{m} / 2 π}{{(ω - ω_{0})}^{2} + γ_{m}^{2} / 4} d ω}{\int^{​} g (ω) \frac{γ_{m} / 2 π}{{(ω - ω_{0})}^{2} + γ_{m}^{2} / 4} d ω} .

\begin{matrix} \frac{d N_{s p}}{d t} = (g_{m} - γ_{m 0}) N_{s p} + r_{s p, m} \\ \frac{d N_{c}}{d t} = R_{e x} - g_{m} N_{s p} - r_{s p, m} - r_{s p, f s} - r_{A} \end{matrix}

R_{s p} = γ_{m 0} N_{s p} .

R_{o u t} = γ_{r a d} N_{s p} = γ_{r a d} / (γ_{r a d} + γ_{n r a d}) R_{s p},

N_{s p} = \frac{R_{e x}}{γ_{m 0}}

R_{s p} = R_{e x}

g_{m} = \frac{γ_{m 0}}{2}

N_{s p, t} = \frac{r_{s p, m}}{γ_{m 0} - g_{m}} \approx {\bar{n}}_{s p, m} \approx 1

\frac{d g_{m} (ω)}{d ω} = g (ω) \frac{γ_{m} / 2 π}{{(ω - ω_{0})}^{2} + γ_{m}^{2} / 4}